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Mirrors > Home > MPE Home > Th. List > vccl | Structured version Visualization version GIF version |
Description: Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vciOLD.1 | ⊢ 𝐺 = (1st ‘𝑊) |
vciOLD.2 | ⊢ 𝑆 = (2nd ‘𝑊) |
vciOLD.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
vccl | ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vciOLD.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | vciOLD.2 | . . 3 ⊢ 𝑆 = (2nd ‘𝑊) | |
3 | vciOLD.3 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | vcsm 30495 | . 2 ⊢ (𝑊 ∈ CVecOLD → 𝑆:(ℂ × 𝑋)⟶𝑋) |
5 | fovcdm 7596 | . 2 ⊢ ((𝑆:(ℂ × 𝑋)⟶𝑋 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) | |
6 | 4, 5 | syl3an1 1160 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 × cxp 5680 ran crn 5683 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 1st c1st 8001 2nd c2nd 8002 ℂcc 11156 CVecOLDcvc 30491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-ov 7427 df-1st 8003 df-2nd 8004 df-vc 30492 |
This theorem is referenced by: vc0 30507 vcm 30509 nvscl 30559 |
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